Distributing Retractions, Weak Distributive Laws and Applications to Monads of Hyperspaces, Continuous Valuations and Measures

Given two monads $S$, $T$ on a category where idempotents split, and a weak distributive law between them, one can build a combined monad $U$. Making explicit what this monad $U$ is requires some effort. When we already have an idea what $U$ should be, we show how to recognize that $U$ is indeed the combined monad obtained from $S$ and $T$: it suffices to exhibit what we call a distributing retraction of $ST$ onto $U$. We show that distributing retractions and weak distributive laws are in one-to-one correspondence, in a 2-categorical setting. We give three applications, where $S$ is the Smyth, Hoare or Plotkin hyperspace monad, $T$ is a monad of continuous valuations, and $U$ is a monad of previsions or of forks, depending on the case. As a byproduct, this allows us to describe the algebras of monads of superlinear, resp. sublinear previsions. In the category of compact Hausdorff spaces, the Plotkin hyperspace monad is sometimes known as the Vietoris monad, the monad of probability valuations coincides with the Radon monad, and we infer that the associated combined monad is the monad of normalized forks.